= Damping matrix
{title2=$C$}
A \b[damping matrix] $C$ maps generalized velocity to the velocity-dependent resisting-force term in a linear <structural dynamics> model:
$$
M\ddot u+C\dot u+Ku=f(t).
$$
Here $M$ is the <mass matrix> and $K$ is the <stiffness matrix>. The sign convention places $C\dot u$ on the left, so the actual resisting force is $-C\dot u$. For pure translational coordinates, entries have units $\mathrm{N\,s\,m^{-1}}$; mixed rotational and translational coordinates require their corresponding generalized units.
Consider an original two-coordinate model with three <viscous dampers>: one from each coordinate to a fixed support, and a third between them. With positive coefficients $c_1,c_2,c_{12}$,
$$
C=\begin{pmatrix}c_1+c_{12}&-c_{12}\\-c_{12}&c_2+c_{12}\end{pmatrix}.
$$
For $\dot u=(v_1,v_2)^T$, the removed mechanical <power> is
$$
\dot u^TC\dot u=c_1v_1^2+c_2v_2^2+c_{12}(v_1-v_2)^2\geq0.
$$
This makes $C$ a <positive semidefinite matrix>. In particular, the connecting damper dissipates nothing when both coordinates move at the same speed. Choosing $c_1=c_2=10$ and $c_{12}=5$ in newton-seconds per metre gives $C=\left(\begin{smallmatrix}15&-5\\-5&15\end{smallmatrix}\right)$. Speeds $(1,0)\,\mathrm{m\,s^{-1}}$ dissipate15watts, while $(1,1)\,\mathrm{m\,s^{-1}}$ dissipate20watts.
For symmetric constant $M,K$, multiplying the equation by $\dot u^T$ gives an energy balance:
$$
\frac{d}{dt}\left(\frac12\dot u^TM\dot u+\frac12u^TKu\right)=\dot u^Tf-\dot u^TC\dot u.
$$
The matrix therefore encodes where <dissipation> occurs and which relative motions activate it.
<Rayleigh damping> approximates $C$ by a weighted combination of the mass and stiffness matrices. OpenSees exposes separate coefficients for mass and different stiffness states, and cautions that this model can give incorrect results in some nonlinear concentrated-plasticity analyses.<Rayleigh Damping Command in OpenSees Documentation>[Section3.1.13.1, formula and Note]{ref} A convenient matrix prescription must still represent the intended dissipation mechanism.
Back to article page
